60-second revision
- Exterior angle of a triangle = sum of the two interior opposite angles.
- Centroid divides each median 2:1 from the vertex.
- Right triangle: inradius = (a + b − c)/2, circumradius = half the hypotenuse.
- Similar triangles: area ratio = square of the side ratio.
- Regular n-gon: exterior angle 360/n; diagonals n(n − 3)/2.
- Angle at the centre = 2 × angle at the circumference on the same arc; angle in a semicircle = 90°.
- Opposite angles of a cyclic quadrilateral add to 180°; tangent length = sqrt(d² − r²).
Triangle basics
- The three angles add to 180°. An exterior angle equals the sum of the two interior opposite angles.
- Any two sides together are longer than the third; the larger side faces the larger angle.
- Angles in ratio a:b:c are 180 × a/(a + b + c) and so on.
The four centres
| Centre | Where the lines meet | Key fact |
|---|---|---|
| Centroid G | Medians | Divides each median 2:1 from the vertex |
| Incentre I | Angle bisectors | Angle BIC = 90° + A/2; centre of the incircle |
| Circumcentre O | Perpendicular bisectors of sides | Angle BOC = 2A (acute triangle); midpoint of hypotenuse in a right triangle |
| Orthocentre H | Altitudes | Angle BHC = 180° − A; at the right-angle vertex in a right triangle |
Worked example: if angle A = 70°, then angle BIC = 90 + 35 = 125°, angle BOC = 140° and angle BHC = 110°.
Right triangles
Pythagoras: hypotenuse² = sum of the squares of the other two sides. Learn the triplets 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41 and their multiples (6-8-10, 9-12-15).
For legs a, b and hypotenuse c: inradius r = (a + b − c)/2 and circumradius R = c/2. Worked example: for 5-12-13, r = 2 and R = 6.5.
Congruence and similarity
- Congruence rules: SSS, SAS, ASA, AAS and RHS. AAA proves only similarity; SSA proves nothing.
- Similarity rules: AA, SSS (sides in proportion), SAS (two sides in proportion with the included angle equal).
- In similar triangles, sides, perimeters, medians and altitudes are in the same ratio; areas are in the square of that ratio.
- Basic proportionality theorem: a line parallel to one side divides the other two sides in the same ratio.
- Angle bisector theorem: the bisector of angle A divides BC in the ratio AB:AC.
Worked example: AB = 6, AC = 9, BC = 10. The bisector of A cuts BC in the ratio 6:9 = 2:3, so BD = 4 and DC = 6.
Worked example: two similar triangles have corresponding sides 3 cm and 7 cm; their areas are in the ratio 9:49.
Polygons
| Quantity | Formula | Regular hexagon (n = 6) |
|---|---|---|
| Sum of interior angles | (n − 2) × 180° | 720° |
| Each interior angle | (n − 2) × 180°/n | 120° |
| Each exterior angle | 360°/n | 60° |
| Number of diagonals | n(n − 3)/2 | 9 |
Circle theorems
- The angle at the centre is twice the angle at the circumference on the same arc; angles in the same segment are equal.
- The angle in a semicircle is 90°.
- Cyclic quadrilateral: opposite angles add to 180°; an exterior angle equals the interior opposite angle.
- A tangent is perpendicular to the radius at the point of contact; two tangents from an outside point are equal.
- Tangent length from a point at distance d from the centre = sqrt(d² − r²).
- The perpendicular from the centre bisects a chord: chord = 2 sqrt(r² − p²), where p is its distance from the centre.
- Two circles with centres d apart: direct common tangent = sqrt(d² − (r1 − r2)²); transverse = sqrt(d² − (r1 + r2)²).
Worked example: in a circle of radius 10 cm, a chord 6 cm from the centre has half-length sqrt(100 − 36) = 8, so the chord is 16 cm.
Worked example: from a point 17 cm from the centre of a circle of radius 8 cm, the tangent is sqrt(289 − 64) = 15 cm.
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